To make the boba dough, in a pot over low heat, combine the water and the Taiwanese brown sugar and stir until dissolved. Bring to a full boil.
In a bowl, add the tapioca starch. Carefully pour in the boiling sugar liquid and mix with a utensil. Knead while hot until a dough forms. Transfer to a flat surface and knead until smooth.
To shape the pearls, pinch off small pieces of the dough and roll into small balls. Lightly dust with the extra tapioca starch to prevent sticking.
To cook the boba, in a large pot of boiling water, add the pearls and stir constantly for 1 minute. Reduce to medium heat and cook for 20 minutes until soft and chewy. Turn off the heat, cover, and let sit for 5 minutes.
Drain and transfer the cooked pearls to a bowl of ice water briefly.
To make the brown sugar syrup, in a pot over low heat, combine the water and the Taiwanese brown sugar. Stir to dissolve and bring to a boil. Add the drained boba and simmer for 5 minutes until glossy and darkened.
To assemble, spoon the boba and syrup into a glass. Tilt and swirl the glass so the syrup coats the sides. Add ice, then pour in the milk. Stir before drinking.